Summary
The structure of tensor representations of the classical finite-dimensional Lie algebras was described by H. Weyl.In this paper we extend Weyl’s results to the classical infinite-dimensional locally finite Lie algebras \({\mathfrak{g}{{\mathfrak{l}}}_{\infty}},\,\,{\mathfrak{s}{{\mathfrak{{l}}}_{\infty}}},\,\,{\mathfrak{s}{{\mathfrak{{p}}}_{\infty}}}\,\,\,{\rm and}\,\,\, {\mathfrak{s}{{\mathfrak{{o}}}_{\infty}}},\) and study important new features specific to the infinite-dimensional setting. Let \(\mathfrak{g}\) be one of the above locally finite Lie algebras and let v be the natural representation of \(\mathfrak{g}.\) The tensor representations of \(\mathfrak{g}.\) have the form V ⊗p ⊗ V ⊗q * for the cases \({\mathfrak{g}}=\,\,{\mathfrak{g}}{{\mathfrak{{l}}}}_{\infty},\,\,{\mathfrak{s}{{\mathfrak{{l}}}_{\infty}}},\) where V * is the restricted dual of V. In contrast with the finite-dimensional case, these tensor representations are not semisimple. We explicitly describe their Jordan’Höllder constituents, socle filtrations, and indecomposable direct summands.
2000 Mathematics Subject Classifications: Primary 17B65, 17B10.
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Penkov, I., Styrkas, K. (2011). Tensor Representations of Classical Locally Finite Lie Algebras. In: Neeb, KH., Pianzola, A. (eds) Developments and Trends in Infinite-Dimensional Lie Theory. Progress in Mathematics, vol 288. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4741-4_4
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DOI: https://doi.org/10.1007/978-0-8176-4741-4_4
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